Congestion Games with Multisets of Resources and Applications in Synthesis

Congestion Games with Multisets of Resources and Applications in Synthesis Guy Avni1, Orna Kupferman1, and Tami Tamir2 1 School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel 2 School of Computer Science, The Interdisciplinary Center, Israel Abstract In classical congestion games, players’ strategies are subsets of resources. We introduce and study multiset congestion games, where players’ strategies are multisets of resources. Thus, in each strategy a player may need to use each resource a different number of times, and his cost for using the resource depends on the load that he and the other players generate on the resource. Beyond the theoretical interest in examining the effect of a repeated use of resources, our study enables better understanding of non-cooperative systems and environments whose behavior is not covered by previously studied models. Indeed, congestion games with multiset-strategies arise, for example, in production planing and network formation with tasks that are more involved than reachability. We study in detail the application of synthesis from component libraries: different users synthesize systems by gluing together components from a component library. A component may be used in several systems and may be used several times in a system. The performance of a component and hence the system’s quality depends on the load on it. Our results reveal how the richer setting of multisets congestion games affects the stability and equilibrium efficiency compared to standard congestion games. In particular, while we present very simple instances with no pure Nash equilibrium and prove tighter and simpler lower bounds for equilibrium inefficiency, we are also able to show that some of the positive results known for affine and weighted congestion games apply to the richer setting of multisets. 1998 ACM Subject Classification F.2.0 Analysis of algorithms and problem complexity – Gen- eral, J.4 Social and behavioral sciences – Economics Keywords and phrases Congestion games, Multiset strategies, Equilibrium existence and com- putation, Equilibrium inefficiency Digital Object Identifier 10.4230/LIPIcs.FSTTCS.2015.365 1 Introduction Congestion games model non-cooperative resource sharing among selfish players. Resources may be shared by the players and the cost of using a resource increases with the load on it. Such a cost paradigm models settings where high congestion corresponds to lower quality of service or higher delay. Formally, each resource e is associated with an increasing latency function fe : IN → IR, where fe(`) is the cost of a single use of e when it has load `. Previous work on congestion games assumes that players’ strategies are subsets of resources, as is the case in many applications, most notably routing and network design. For example, in the setting of networks, players have reachability objectives and strategies are subsets of edges, each inducing a simple path from the source to the target [29, 3, 19]. We introduce and study multiset games, where players’ strategies are multisets of resources. Thus, a player may need a resource multiple times – depending on the specific resource and strategy, and © Guy Avni, Orna Kupferman, and Tami Tamir; licensed under Creative Commons License CC-BY 35th IARCS Annual Conf. Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Editors: Prahladh Harsha and G. Ramalingam; pp. 365–379 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 366 Congestion Games with Multisets of Resources and Applications in Synthesis his cost for using the resource depends on the load that he and the other players generate on it. Formally, in multiset congestion games (MCGs, for short), a player that uses j times a resource e that is used ` times by all players together, pays j · fe(`) for these uses. Beyond the theoretical interest in examining the effect of multisets on the extensively- studied classical games, multiset congestion games arise naturally in many applications and environments. The use of multisets enables the specification of rich settings that cannot be specified by means of subsets. We give here several examples. As a first example, consider network formation. In addition to reachability tasks, which involve simple paths (and hence, subsets of resources), researchers have studied tasks whose satisfaction may involve paths that are not simple. For example, a user may want to specify that each traversal of a low-security channel is followed by a visit to a check-sum node. A well-studied class of tasks that involve paths that need not be simple are these associated with a specific length, such as patrols in a geographical region. Several communication protocols are based on the fact that a message must pass a pre-defined length before reaching its destination, either for security reasons (e.g., in Onion routing, where the message is encrypted in layers [27]) or for marketing purposes (e.g., advertisement spread in social networks). In addition, tasks of a pre-defined length are the components of proof-of-work protocols that are used to deter denial of service attacks and other service abuses such as spam (e.g., [15]), and of several protocols for sensor networks [7]. The introduction of multiset corresponds to strategies that are not necessarily simple paths [5]. In production systems or in planning, a system is modeled by a network whose nodes correspond to configurations and whose edges correspond to actions performed by resources. Users have tasks, that need to be fulfilled by taking sequences of actions. This setting corresponds to an MCG in which the strategies of the players are multisets of actions that fulfill their tasks, which indeed often involve repeated execution of actions [13]; for example “once the arm is up, do not put it down until the block is placed". Also, multiset games can model preemptive scheduling, where the processing of a job may split in several feasible ways among a set of machines. Our last example, which we are going to study in detail, is synthesis form component libraries. A central problem in formal methods is synthesis [26], namely the automated construction of a system from its specification. In real life, hardware and software systems are rarely constructed from scratch. Rather, a system is typically constructed from a library of components by gluing components from a library (allowing multiple uses) [23]. For example, when designing an internet browser, a designer does not implement the TCP protocol but uses existing implementations as black boxes. The library of components is used by multiple users simultaneously, and the usages are associated with costs. The usage cost can either decrease with load (e.g., when the cost of a component represents its construction price, the users of a component share this price) as was studied in [4], or increase with load (e.g., when the components are processors and a higher load means slower performance). The later scenario induces an instance of an MCG. Let us demonstrate the intricacy of the multiset setting with the question of the existence of a pure Nash equilibrium (PNE). That is, whether each instance of the game has a profile of pure strategies that constitutes a PNE – a profile such that no player can decrease his cost by unilaterally deviating from his current strategy. By [28], classical congestion games are potential games and thus always have a PNE. Moreover, by [19], in a symmetric congestion game, a PNE can be found in polynomial time. As we show in Example 1 below, a PNE might not exist in an MCG even in a symmetric two-player game over identical resources. G. Avni, O. Kupferman, and T. Tamir 367 Table 1 Players costs. Each entry describes the cost of Player 1 followed by the cost of Player 2. {a, a, b} {b, b, c} {c, c, a} {a, a, b} 36, 36 19, 17 17, 19 {b, b, c} 17, 19 36, 36 19, 17 {c, c, a} 19, 17 17, 19 36, 36 Example 1: Consider the following symmetric MCG with two players and three resources: a, b, and c. The players’ strategy space is {a, a, b} or {b, b, c} or {c, c, a}. That is, a player needs to access some resource twice and the (cyclically) consequent resource once. The 2 latency function of all three resources is the same, specifically, fa(`) = fb(`) = fc(`) = ` . The players’ costs in all possible profiles are given in Table 1. We show that no PNE exists in this game. Assume first that the two players select distinct strategies, w.l.o.g. {a, a, b} and {b, b, c}. In this profile, a is accessed twice, b is accessed three times, and c is accessed once. Thus, every access of a, b and c costs 4, 9 and 1 respectively. The cost of Player 1 is 8 + 9 = 17, while the cost of Player 2 is 18 + 1 = 19. By deviating to {c, c, a}, the cost of Player 2 will reduce to 17 (while the cost of Player 1 will increase to 19). Thus, no PNE in which the players select different strategies exists. If the player select the the same strategy, then one resource is accessed 4 times, and one resource is accessed twice, implying that the cost of both players is 2 · 16 + 1 · 4 = 36, and any deviation is profitable. We conclude that no PNE exists in the game. We study and answer the following questions in general and for various classes of multiset congestion games (for formal definitions, see Section 2): (i) Existence of a PNE. (ii) An analysis of equilibrium inefficiency.A social optimum (SO) of the game is a profile that minimizes the total cost of the players; thus, the one obtained when the players obey some centralized authority.

Load more